Photoelectric absorption in semiconductors

[MicheleC88]

Hi everybody. I'm new here and, first of all, sorry for my bad english :-D

I'm studying photoelectric absorption in semiconductors.
The book (and professor too) says that, in the conservation law:

k_i + k_ph = k_f

(where k_i and k_f are wave vectors of initial and final electron state, and k_ph is the wave vector of incoming photon) we can neglect k_ph because it is ≈ 2π/λ, whereas k_i and k_f are ≈ 2π/a, and λ>>a. (a is the length of unitary cell in real space).
But I think that this assumption is good only if electron is at the edge of Brillouin Zone; if the initial and final electrons are near \Gamma-point, they should have a very little wave vector, comparable with k_ph, making the approximation not valid.

what is wrong in my words?

 

[Cthugha]

Just do the math once and sketch the photon dispersion (energy versus k) and the electron dispersion (crystal electron or for simplicity even a free one) into the same graph with the correct dimensions. This is pretty instructive and will give you a good argument for why the approximation your professor gave you is a very good one pretty much everywhere.

 

[MicheleC88]

So, I think having understood from your words, the key is that the conservation of momentum has to be combined with the conservation of energy?

 

[ZapperZ]

Actually, that conservation law is valid only for the in-plane momentum, i.e. parallel to the surface of the material. The out-of-plane momentum is way more complicated than that.

Zz.

 

[MicheleC88]

I tried to make the following math.
from energy conservation: E_ph≈E_gap
(assuming that initial and final electrons are in proximity of, respectively, the maximum of VB and minimum of CB)

We know that k_ph = ω/c = E_ph/(\hbarc)

So, from momentum conservation law:

|k_f - k_i | = |k_ph|≈E_gap/(\hbarc)

Doing the calc (i assumed 1 eV for E_gap):
|k_f - k_i | ≈ 10^-4 angstrom^-1, which is about 1 part of thousand of tipical size of Brillouin zone.

Are my reasoning correct to justify the assumpion k_ph=0 ?

 

[MicheleC88]

However, I think I made the math more complex than necessary:

k_i - k_f = k_ph = \frac{2π}{λ}\widehat{k}_{ph}

(k_i - k_f) / (size of Brillouin zone) = \frac{2π}{λ}\frac{a}{2π}\widehat{k}_{ph} << 1 \cdot \widehat{k}_{ph}

Thanks to all for the reply!